081 = Pi Prize and Square Stairs

Hello and welcome to A Problem Squared, the problem-solving podcast, which is a lot like Pi in that we have a habit of going on forever.

So true.

But unlike Pi, we also have a habit of repeating ourselves.

So true.

Your hosts are myself, Beck Kill, and Matt Parker.

That's me.

And Matt, you are also like Pi.

Oh, yeah.

Because you are maths related.

Correct.

And have a very large but niche fan base.

Ha.

Yeah.

And I think it overlaps a lot with Pi's fan base.

Yeah,

I think there is a...

You're in the Venn diagram, it's essentially a circle.

Appropriate.

And I am also like Pi because I'm full of chunks of meat and sometimes a little crusty.

So close.

A little crusty.

A little crusty in the mornings.

Oh, fair enough.

I have no follow-on questions.

In this episode, I work out if pie can almost repeat itself.

I'll be staring at some stares.

And we got any other business about me doing unnecessary detail in New York on the 14th of April.

Ha!

I got it in the menu as well.

Damn it.

You didn't want to talk about Pi Day?

Oh, and that, and that.

Hey, Matt.

Back.

Well, I mean, now that I've mentioned it in the menu,

Pi Day.

I'm guessing, because this episode comes out just before Pi Day.

Just before Pi Day.

Monday before Pi Day.

Yes.

Very exciting.

Yeah, I've noticed that here we celebrate Pi Day based on the American date of the Spanish.

We do.

Only because we haven't got a better suggestion.

Right.

So if we had a 31st of April, which we would write as 314,

we'd have a superior Pi Day.

But we don't.

That date doesn't exist.

Yeah, that's true.

All we get is the 14th of March, which you have to write the wrong way around to get 314.

but I'm not going to pass up an opportunity to celebrate pie.

What about the 3rd of February, which is the 14th month if you can't roll over?

No, I like what you're doing.

That's Valentine's Day.

Oh, Valentine's Day is Pie Day.

About what Pie Day is.

No, I'm all right, everyone.

I want everyone to understand that I am now, I'm starting a rival Pie Day on Valentine's Day.

You've all got 11 months to get ready for it.

Yes.

Because what is a love heart if not a circle with a point?

That's That's what they say.

Two points, technically.

I think it was Euclid who said that.

Yeah.

So it's Pi Day.

I celebrate, well, I have now for almost a decade celebrated every Pi Day by calculating Pi a ridiculous way.

Yes.

Which I kind of started doing by accident.

And it just so happened the first time I did it.

Sorry, how did you do it by accident?

Well, I always liked calculating pie silly ways.

And then when I first started doing my YouTube channel, one of the first videos I made, because I started at kind of the beginning of the year and then Pi Day came up.

This was my original mini oit.

I was like, oh, I should do a thing about pi.

And so I calculated pi by weighing a circle to work out the value of pi.

Oh, yes, yeah.

Which I thought was very funny.

And I didn't think anything else of it until the next Pi Day rolled around a year later, still doing YouTube.

And I'm like, oh, what's another ridiculous way?

And I quite like there are some infinite series where you're adding or subtracting like infinitely many terms, but the more you do, the closer you get to pi.

So you get more digits as you do more of them.

And there's a terrible one that I did.

And so I thought it would be kind of funny to do, worked it out in a big chalkboard, vaguely pie-ish.

Job done.

What I didn't realize is I'd now set myself up into a rhythm where every year I calculate pi, every odd year I do some kind of ridiculous experiment

to physically calculate pi.

And on the in-between even years, I try and work it out by hand.

Yep.

And my working it up by hand gradually escalated.

The next year, or two years later, I then did a better calculation, but it took me all day to try and get more digits.

And then two years after that, I got five people in a room to help.

And then two years after that, I had 20, 20 to 30 people in a room for a weekend for two days to help.

And then two years after that, which brings us up to this year, I had 200 people in a room

for six days calculating pie by hand.

Oh, my goodness.

It's kind of got out of hand in that regard.

Yeah.

Yeah.

We went all out.

Oh my goodness.

I'm nervous thinking about 2026.

Oh, yes.

We'll get to that.

So our goal was...

So the current world record, if I may.

Yeah, you're using quotation marks.

Because no one's ever contested this record.

Because the record was set before

we had electronic computers to calculate pi.

Oh, okay.

We're not 100%

sure if the person who did this, William Shanks, who calculated pi on and off between about 1850 and the 1870s,

if they had any kind of mechanical calculator to help them.

We don't know.

So for now, we're assuming they did it all by hand until we have any evidence otherwise.

They thought they had 707 digits correct.

Nobody double-checked.

In the 1940s, someone did double-check, now with a mechanical calculator, and they realized they made a mistake in the 528th digit.

Oh.

So they put the effort in for 707, and 527 were correct.

I really want to beat that.

I mean you don't want to cross it.

I want to have the record.

William, because I've heard that William Shanks.

Shanks.

William Shanks.

Oh, Billy Shanks.

I know you've all made.

Yeah, you've made all the jokes.

It's great.

So, yeah, so that's the record to beat.

That's the record to beat.

And it might be less than that, but hey, why not have a benchmark?

If we beat that, we would unambiguously have the world record for most digits.

And is that what you were going for?

We started with the capacity.

If everything went perfectly, we were in with a vague shot.

Can you explain to me how you calculate pi by hand?

Great question.

So there are a bunch of different equations that give you pi as the answer, but some of them are quite complicated.

They contain lots of summation, which is

where you're adding lots of things together.

Okay.

Some of them involve multiplying things together.

Some of them involve factorials, which is those are the numbers where you're multiplying by every number smaller than them.

And the reason that they're not good is they get very big very quickly.

Okay, some of them involve square roots or other kind of roots, which are very difficult to do by hand.

Some involve raising things to weird powers, like non-integer powers, which are difficult to do by hand, and some of them involve dividing very big numbers by each other, which is hard to do by hand.

And the reason we have all these complicated equations is because most of the time, mathematicians only care about is it a cool equation that gives you pi?

And

the actual practical ones are optimized for one definition of practical, are optimized for computers to do.

They're not optimized for humans to do.

So I had to go through all these different equations and try and find one which could be done, first of all, could be done by humans by hand.

Secondly, it could be split up between a lot of humans to all do at once.

Yeah, because that's where my, like where I sort of get stuck, because I'm like, how do you

divide up the

sum?

Yeah, so first of all, we picked an equation.

I say we, me and the council of calculating pie by hand we picked an equation which is actually seven different equations and at the end you've just got to add their results together.

Okay.

So each one can be done separately.

So already now we've got seven different things that can be done independent of each other.

Within each one it's actually splits into a whole bunch of things you've got to add together and you can kind of once you've started like let's say you've calculated the first one you've done the first 20 digits of it.

We were doing them in 20 digit chunks.

So the first person does the long division to get the first 20 digits.

They can then hand that on to the next person who uses it to get the first 20 digits of the next term, who then hands it on to the next person to get the first 20 digits of the next term.

But meanwhile, the people who do the first 20 digits of the first term, they've now done 20 to 40 digits, then they've done 40 to 60, and they're working their way down.

So you've got some people who are doing digits further and further into one of the many, many different things we need to know.

And other people are taking what's already been done and starting on the

subsequent things we have to do.

Yeah, my brain is sort of imagining like a family tree, like a sort of flow tree.

Very much like a family tree.

Yeah.

But it's weird because depending on what you're doing, so we call each sheet as a chunk.

So, so the goal was in terms of the end user, we had, oh goodness,

somewhere between three and four hundred unique humans came and helped out at some point in time.

Yeah.

For each human calculator, all they have to do is walk up to one of the handing out tables we had,

get a sheet, and all the sheets are long division.

They'll have a big number at the top, they'll have a smaller number they have to divide into it, and they go away and do that long division, accurate to 20 digits, and then they bring and hand it back in again.

So from the end user's point of view, you're just doing lots of long division.

From our point of view, running the machine, each one, like each sheet that's done, the things like the answer has to go to two different places to two different sheets.

But some of the sheets require two different bits of information to come in before they can be sent out to be divided.

And the logistics behind tracking all of these sheets yeah and making sure the right things are copied into the right place with no mistake because we can't a single mistake and we're still this was i was going to say is there like a quality control element is there anyone that checks someone else's sheet after they've done it so we sent out each sheet three times independently

waited until two came back that matched yeah that was the first verification that was done by the verification station they would then pass it on to the mod squad who would do a modular arithmetic check, which is where you do, like you can't use it to get the answer in the first place, but you can redo the calculation.

They were doing it, for people who are familiar with these things, modular 9 and modular 11.

And I mean, mod 10, you kind of get for free.

And then they would compare the results from those to make sure it lined up with the answer that was on the sheet.

So it's a way of doing a real shortcut version of the calculation.

but it will indicate if the one on the page is probably correct or not.

By doing that, we found on one of the long divisions, two different people doing the same division both made the same mistake at exactly the same place.

And that meant it passed the verification check of comparing their answers.

Their answers are identical, but they just accidentally made the same mistake at the same time.

And if you're doing, we were doing thousands of sheets.

Yeah, I was going to say that's like,

yeah.

Yeah.

But the mod squad caught it because it didn't pass the mod test.

We estimate the mod squad would catch 90% of mistakes that got to it.

Wow.

Which is good,

but

it still leaves 10% chance.

And if we were to do this again for even more digits.

So the video came out today on the 11th of March when this podcast goes out.

Okay.

So.

So the listeners know something that I don't.

Potentially.

But I don't want to ruin the surprise if you haven't seen the video yet.

So I'm going to give away most of the result.

Okay.

We, at the end of six days with a lot of people, had calculated 140 digits of pi.

Calculated, not checked.

Okay.

We then had to check that against the real value of pi.

Last time we did this, we got 11.

11 digits correct.

Oh.

We want more than that.

Wait, hang on.

When you say 11 digits correct,

do you mean like in the places they're supposed to be?

In the place where they're supposed to be, yes, yes, yes.

What, like the first 11 digits?

Yeah, yeah, at the beginning, we got 3.1415926 and so on for the first 11 digits.

And then we had a mistake.

And then the rest of it is

wrong.

So you can.

So that we only had like 30 people for two days then.

Yeah, but you had 30 people for two days and you calculated by

stop it.

I know.

I know.

Look.

No, no, no.

You're late to the party.

A lot of people have already found that very funny.

I know.

But I think you deserve your moment to appreciate it.

Thank you.

Yeah.

It's very funny.

Your response is valid.

But I would also say that, I mean, you know me.

I'm always about going the long, fun way even if it's wrong oh yeah because it's more fun yes so I applaud and I appreciate that effort so it would have been very embarrassing to get hundreds of people together for a week

only 11 correct

so I can reveal for our listeners that we made over 100 digits

Okay, that's impressive.

It is astounding.

Even if it's 101, I'm not going to be able to do that.

I'm not going to tell you how many over we got it's still impressive we locked in over a hundred digits correct i'll be honest believe it with the because i don't understand what what like you've even though you've explained to me what goes into calculating it oh it's a whole thing i i'm i'm still like i thought you would have gotten way over 500 and whatever

you've got to see the chaos that led to i also forget like what it look if i was to write out like as if I was targeting

500 and something that in itself is like long long.

Never mind doing the maths.

For me, it was a real experience.

Like, you think you know pie, but watching that chaos, and yet you get these exact digits out the other side.

Like, why?

Why on earth should that ridiculously

translation?

I was very emotional.

It's pretty special beginning to end.

Awesome.

So.

And how have you been back?

I'm good.

I'm good.

Speaking of YouTube videos, you finally watched mine.

I did.

Yeah.

I did, of my own free will.

Yes.

It's not like you came around to my house and sat me down and said, watch the video.

I was like, Matt, it's six minutes long.

Oh, that did happen.

And you're in it.

I mean, I consume a lot of YouTube, big YouTube fan, as well as making it.

You've done what a lot of people on YouTube don't do, and you've put thought and structure into the whole video as a whole.

Well, which I really like.

You could say that I

was inspired.

You work as an actual comedy writer.

It's a good starting place, I figure.

Oh, yeah, yeah, yeah.

Excellent start to yoit.

The year of YouTube.

Thank you.

I was mainly just wanting to make sure that you were happy with your cameo.

It's good.

The only thing I regret is I should have given you some B-roll of me buying the snacks in Japan.

Oh, yeah.

That would have been nice.

It's good.

It's short.

No YouTube videos ever finished.

At some point, you just got to draw a line under it and put it on.

I think it was Da Vinci that once said that.

I think it was.

Yeah.

Yeah.

And I do have a little bit of other stuff to say because this comes out on the 11th.

Other than everyone go watch the video?

Yes.

I know that Pi Day is coming up for you, but last night, well, I guess both UK and US time was the Oscars.

Maybe I'm just announcing this after it's already happened and there's nothing anyone can do about it, but I'm going to be doing a little spot on Jonathan Ross is going to be doing a little pre-Oscars companion show before they start streaming live from the States.

And so I'm going to

pick up a flip chart, flip chart, and do some

moving flip chart stuff.

Yeah.

So it should be fun.

But there were some some suggestions that I had.

Like one of them was,

I wanted, let's see if you can guess what this would have been.

Okay.

If I'd drawn it.

It's a picture of Nicholas Cage.

Yep.

And he's got a very big breast on him.

Just the one.

Yeah.

Right.

The breast is put onto him, like as, like a pin the tail on the donkey type thing.

Okay.

So you've got to guess what the film

is.

And I'm going to say it's more of a say what you see.

Don't get thrown by the fact that it's an actor.

No, but you're kind of like, you're so close to it already.

Ah, Titon Nick.

What?

Titanic.

Titanic?

Yeah.

Yeah, so dumb.

Yeah, so dumb.

Beck.

Yeah, the execs didn't like that.

Yeah, how about that?

Vetoed.

How have you made me side with TV execs?

Well, thanks everyone for sticking with us because this was quite a lot of catching up.

That's a lot of catch-up.

I think very fascinating.

Let's Let's hope so.

It's Pi Day.

This first problem is for you, Matt.

Yep.

It's from No Name.

They didn't enter a name.

They went to the problem posing page at propsqued.com.

Yep.

Skipped right over that.

To me, no text fields are compulsory.

No, that's it.

They don't.

That's how we live.

You choose what you want to fill in.

Maybe you left us this problem.

It could be me.

No, I meant the listener.

No, the listener.

You.

Yeah, you.

You.

I mean, I wouldn't put it past you, Matt, because surprise, surprise,

it's about pi.

I'm brand.

So, can numbers like pi almost repeat themselves?

Like, could pi go 3.14 da da da da?

Stop, stuff, stuff.

1, 2, 3, followed by 3.14, da da da da da da da da da da da da da da 1 2 4.

My instincts are telling me no, but I don't really have the math knowledge to back that instinct up.

This came up because I had an idea for a short story where a supercomputer calculating the digits to pi has an existential crisis because pi seemingly starts repeating.

I like that.

Yeah.

Yeah.

I would read a short story about a supercomputer having an existential crisis.

I don't think anything I'm going to say is not going to devalue that.

Yeah.

Great story idea.

Yeah.

And we can all take it now because you have to steal it.

Everyone, go.

No, no, no one steal it.

But if you do write this short story, uh, please let us know.

But in the meantime.

Can pi almost repeat?

I mean, if it's technically infinite, then yes.

Well, I mean, do they want it to start repeating from the beginning or do they just want like a patch that repeats partway through?

Mmm.

And they're subtly different questions.

So you know what?

You know, I'm going to give you

one of my favorite facts about pi.

There's a thing called a pi prime, which is where you start from the beginning of pi with the 3, 1, 4, 1, 5, 9.

A pi.

A pi.

And you keep going, and you see how many digits, if you stop there, everything so far is a prime number.

So it works for one digit because 3 is prime.

Okay.

It works for two digits because 31 is a prime number.

Oh, okay, yeah.

3, 1.

It doesn't work for 3 because 314

is not a prime number.

It's even for a star.

And so the first couple are 3, 1, 4, 1, 5, 9, is a prime number.

Okay.

And it works for the first 38 digits.

That's a big prime number.

And you think, oh, I wonder when the next one's going to be.

The one after that is 16,208 digits.

oh that's a big old that's a big one yeah and the one after that is 47,000 something digits okay but that's a less of a leap than that's true yeah yeah there's a big there's just a big gap because instantly my brain's like ooh any patterns because my brain likes to do that but I don't immediately see any patterns it's just because each next digit becomes less likely to make it prime and so once you're kind of escaped from the small numbers where there are a bunch of primes once your numbers got any kind of length to it, it now becomes really unlikely that each subsequent digit is going to make a prime.

Yep.

Because there's just a smaller percentage of numbers of prime at that size.

Of course.

And so that once you've hit that escape velocity, once you've escaped the small numbers, it's very unlikely it'll stop.

But for the case of primes, likely enough that, you know, once you're in the tens of thousands, hundreds of thousands of digits, you've had so many chances for it to be prime, eventually it will stop and be prime.

How do we apply this this now to repeating?

Will the same digits repeat at any point?

Well, first, I thought I'd just look for repeating sections inside Pi.

So I thought, I wonder, I've got a document on my laptop, which has the first one million digits of Pi.

Is this gonna at some point involve some terrible Python code?

Oh, whoa, Mac.

I don't know if you noticed.

Can I smell that?

It's coming.

Terrible Python code.

I don't know if you noticed this morning after I got back from my bike ride.

Okay, yeah.

I then sat down because you were working at the dining room table.

Yes.

And Lucy was having a call in the front room.

I then sat down at the dining room table while the coffee was being made.

Yep.

And I knocked together some terrible Python code.

Fantastic.

Because I was like, I've already got a million digits of Pi here.

Yep.

For people curious, it's from when we filmed a video years ago for Number File where we printed out the first one million digits of Pi.

And I did a walking tour of Pi.

And I'm like, oh, you can see this number here and this bit.

So

I wrote some code that would take the million digits and I could search for different things.

So I just dusted off that code and rewrote a bunch of it to search for repeated strings in Pi.

Yep.

So the question is, what do you think is the longest repeated section?

So you have

a run of digits and then immediately followed by the same run of digits again.

How long a run do you think you'll get in a million digits of Pi?

I'm going to bring up the results while you're having that.

As in like,

you know, the same five digits of the same or six digit.

Okay.

Yeah.

The longest run in a million.

In a million.

And if there's one thing i've learned it's that a million is bigger than i think it is big having walked we printed them out tiny little digits and it was a mile to walk

no

what is so a million such

piece of paper it was it was a ridiculously long piece of paper was it done on like one of those those old-fashioned ticker ticker machines what are they called no i don't know like the old stock market ticker yes no like what mr burns has no yeah yeah brady who who got a printer for number file found a printer in i think it was i I want to say Belgium somewhere like that and what they used was you can buy very very long rolls of

like packaging brown paper so it's brown paper but it's not plastic but it's slightly waterproof and a bit more robust than you'd expect yeah I know this almost like wax paper yeah it was like wax paper and what they did was they just cut the end 20 centimeters off a massive roll because when you buy them from the factory to go to the packaging plant It's like huge rolls of the stuff.

And they just sliced the 20 centimeters off the end of a roll.

And so now he had a a m one mile long piece of paper that's like 20 or whatever it was centimeters wide.

Printing on that was a non-trivial.

Did you please tell me you did a a video just about the paper?

Yes, Brady did a whole second video about how it was printed.

Okay, good.

That's my kind of...

It had to go through the printer twice because I requested labels.

I said we're going to get so lost in Pi.

Like I wrote a bit of terrible Python code.

We could type, we could look at the pie in front of us, because we put it on a runway at an airport.

We could look at the pie in front of us and type in a few digits and hit go and it would tell, oh, you're at this point in pie.

Okay, yeah.

But I was like, it'd be way easier if we could just label every

10,000th digits or something like that.

All right.

Not like,

you didn't label it like a ruler.

No.

Like every

drags up.

I think we might have done every thousand digits or something, but it was done by running it through the printer twice.

So one print run put all the digits on the bit of paper and the second time through the printer put all the locations through.

Oh man, man, I'm such a sucker for that stuff.

I love it.

Oh, it was a logistical, it was a whole thing.

At the end of the day, everyone kind of packed up into the cars because it was like support cars driving us along pie.

Yeah.

And everyone drove back to the beginning.

And I was like, no, no, no, I want to walk.

And so everyone else drove back.

And I was by myself.

And it wasn't raining, but it was like that kind of weather.

And I could just see pie to the horizon.

And so I just started walking.

And I spent like 20 minutes or whatever it is walking by myself along pie.

It was a religious experience.

I was going to say, that's quite the mental image.

Yeah, really.

And just like walking for so long with these tiny digits one after the other, but they're all so certain.

It's similar to, you know, when we calculated 100 and they matched.

Yeah, I was going to say it feels like another emotional.

A million digits that are all so seemingly random, but so defined.

Ah, incredible.

Yeah.

Anyway, so that was a lot of fun.

A million is a big number.

It's a lot of digits of pie.

Yes, yes.

And your question to me was, how long do I think a patch of numbers would be that that matches another patch of numbers?

And now that I know how long that would be if it was just written out, I don't know.

10.

Well, good guess.

Five.

Ah, damn.

Do you know what I was going to say?

Yeah.

It's a tough one because there's no introduction.

I said five to start with.

I had to say five.

And I was like, oh, I was avoiding saying a number.

Yeah.

So the first batch of five, which occurs at position, so this is digit 107,472-ish.

This is terrible Python code.

Yep.

There's probably an off-by-one error in there.

It's around there.

Within a couple.

The old TPC.

The old terrible Python code.

64015.

That's my phone number.

No, it's fine.

That repeats twice.

So Pi, once you're 107,472 digits in, goes 64015, 64015.

So it repeats for five digits.

64015.

Yeah.

In a million.

And I don't think it gets more likely.

Like, I know we know pi now to tens of trillions of digits.

I suspect we're not getting much bigger.

Well, it would be bigger than that.

But maybe it might only be 10.

Maybe a bit more.

It's not going to be long.

So pi is, we think, what's called a normal number.

And normal in this context in maths means any...

any group of digits are equally likely.

So

if you pick five digits at random, that's as likely to appear in pi as any other group of five digits yeah and we've not proven that but as far as we can tell it's true for pi it's a very hard thing to prove for for numbers imagine that we we got to a point where we like calculated pi to like quadrillion yeah whatever and then and then just afterwards

and then afterwards it repeats from the beginning from the beginning but like obviously like the final number has to be wrong i mean it could there's nothing stopping it from repeating from the beginning but it it won't definitely just keep doing that over and over.

Like, it could do it once.

Yeah.

And then other stuff.

We know that.

We've proven that.

We're very confident in that.

Now, the issue with getting repeating patches is if you've got a run of five digits, which is one of 100,000 options, the chance of the next five being the same is one in 100,000.

Which is why I'm not surprised the first one happened at 107,000 digits in.

Like, that's about right.

Yeah.

So each time you want to have one more digit, it's 10 times less likely to occur.

Okay.

And will probably happen 10 times on average further into pi.

Got it.

So if you, if you get, if you know 10 times as much pi, the string of repeating digits you're likely to find goes up by one.

Right.

So it's it's real real unlikely.

And that's before we get to if you want it to start at the beginning.

Yeah.

Because we're going to have that same problem where once you're a few in, like once you're 100 digits in, and we know it doesn't repeat in the first hundred digits,

the chance of the next one happening is one in 10 to the power of 100, like one in a Google.

Got it.

Yeah.

So if we don't have a repeat of 3, 1, 4, something near the start.

Something, whatever, then,

yeah, what you're saying is we're getting further and further away from the start until that 314 has to happen again.

In which case, yeah, it's going to be,

I imagine.

The probability of the whole next block matching is one over 10 to however many digits we've done so far.

Yeah.

Because also this unnamed person, they haven't necessarily said how long it would need to be to be something that feels like it's repeating itself.

Oh, that's very true.

And they did.

Because you could say it's just two digits.

They did say almost.

So there is some flexibility.

Like, you may be, oh, maybe like one digit in the middle is wrong.

But I don't, that's not going to, it'll gain you a bit, but not a lot.

And it's even worse than the primes because each time you go one longer, because you're thinking, well, sure, it's very unlikely, but pi never stops.

However, the probability is

not like converging on some kind of non-zero value.

My hand-wavy argument, and people can maybe put this on more rigorous footing than me, is it converges to zero.

So the probability tends to zero.

It doesn't matter if you've got infinitely many digits.

It's a zero probability chance of getting that repeat because the chance of it happening gets small so fast.

So some people would argue that if something is infinite, then it increases the possibility of something happening.

Yes.

But however, something being infinite also means that it increases the impossibility of something happening.

Yeah, because what we want to happen depends on how long,

how far we are into quote-unquote infinity so far.

And the getting difficultness as you go infinitely far gets bigger bigger faster.

Yeah, like if it was happeningness.

One out of three, something will happen.

Two out of three, something won't happen.

Yeah.

And if we just would go, okay, let's like expand that.

Let's just like

go make it big.

Then you've still got more chance of it not happening than you do of it happening.

Yes.

But more complicated than that.

More complicated than that.

Yes, yes, yes, yes.

But this is the first step my brain is taking to sort of vaguely understand what you're saying.

Great, great zero step.

But the problem is both probability and things tending to infinity are

famously woolly and complicated things in mathematics.

But to a very simple and I'm hand waiting.

To a very simple-minded person such as yourself.

Such as me.

No, I'm just saying very, very hand-wavy.

Yes.

This is the vague sense of what's going on.

Yeah.

Yeah.

Yeah.

Like how emojis don't really

tell you exactly how you feel.

You're feeling.

Yeah.

But

they're a zero-step.

They're better than the null message.

I think you've absolutely answered that.

Thank you.

And I think that this no-name person has.

I mean, it's a great concept for a story because I've almost had an existential crisis.

I'm good to go.

I'm trying to wrap my head around this.

I love it.

I like that story.

Yeah.

That's a great answer.

Thanks, Matt.

I'm going to give that a ding, ding, ding, ding, ding, ding, because it's repeating itself.

Ah, the repeating dings.

Next problem.

Put into the problem posing page.

And they did enter a name.

Jelly.

Jell.

Or yell.

Or yell.

Or yelly.

Oh,

let's go with yell.

So, this is yell.

Or hell.

Hell.

Depending on where they're from.

There's a J.

It starts with a J.

It starts with a J.

J-E-L-L-E.

says that they have a solid 1 meter by 1 meter by 1 meter cube

unknown substance.

And they want to to cut it.

Okay, so this I think is a confusing way to put it.

They want to cut it around the diagonal of the XY plane.

Let's not worry about that for now.

Okay.

They want to cut the cube.

That's what we're taking away from this.

Right.

To produce two sets of stairs.

Or in other words, this is Jell speaking again.

They want to build two sets of stairs to use in a stage play, so I guess like a fancy prop, such that one stair, I guess a set of stairs, can be put on the other one to form a perfect cube.

Also, the stairs should be walkable on by an adult human.

Well, that depends what the cube's made of, Jel.

Yes, I guess.

But they want to have two sets of stairs such that if you stack one on top of the other, these two identical sets of stairs form a complete cube.

Yep.

Okay.

We got it.

They say that after a few paper experiments, it seems to be impossible to produce two identical stairs.

One always seems to have more stairs.

This is a good problem.

It's deemed impossible.

That's where we we come in.

They want to know, is it really impossible for stairs to be identical?

If so, what shape of stairs could be close to identical?

And they then go on, blah, blah, blah.

Thanks for an awesome podcast.

If people want to have a go at solving this themselves, now is a good time to pause, give it a go, because, Beck, have you got a solution for us?

Yes.

So

I love this because this is like a puzzle to me and I love puzzles.

And so, you know, your first instinct is to draw a square that represents a cube and then you sort of

draw some steps yeah and when you're drawing the steps you know if you're sort of doing them like cube shaped or whatever you do find that if you go from like the like one step up yeah and then to the and then across and then up again another step and across and if you're doing that in sort of even cubes yep even uh squares then what happens is you come out the diagonal top of your square on the like the same side of that diagonal.

Does this make sense?

Only because I did get a bit of paper and try and do this

all the problem.

If you're able to, you should do it right.

You should try it right now.

But essentially what it means is you end up with a smaller side.

Yes.

You've got one side that's got...

So let's say I've got a three by three

square.

Yep.

That first square is your first step.

First step.

Next two squares.

Yep.

That's your next step up.

And then the next three squares

going upwards.

Three steps.

That's your final step But then the other bit left over you've only got three tiny bits.

You've only got like one step two step up and then yeah, so then

how embarrassing but I was like that well that's weird

I guarantee the first time you try this if you don't put any thought into it in advance you will draw asymmetric stairs.

Yes.

Yeah.

And I I'd had a I realized I'd had a similar problem to this because in a flip chart that I recently made and filmed.

Actually, this one is on you.

This one's on YouTube.

Yeah.

Yeah.

It's called this page is is the First and Also the Last.

On one of the pages, there's a like a broken heart.

And when I was drawing the broken heart, I was trying to do like the break in the middle.

Yeah.

And I was trying to do it so that they had like an even number of

pointy bits.

But that was really hard.

Same problem.

Same problem.

And so I was like, oh, I've come across this before.

I was sitting with you when this problem came in and I was like, this doesn't seem right.

I feel like there is definitely an answer.

And we had very different ways of

doing it.

And

my way was hands-on.

I like your ways.

So I was like, do you have

anything cube-shaped?

Do you have a bunch of dye?

I went and got the dice box from upstairs.

Yes.

I was like, wait here.

And I ran up.

I came back with a box of dice.

Yes.

Most of which are not cubes, but enough were.

No.

And then

I instinctively went to do a cube of three by three by three.

Yep.

I don't know why.

That's just always my instinct.

Three by three.

That's a classic cube.

Yeah.

You obsessed with the Rubik's Cube.

That's true.

Do you know what?

I was playing with the Rubik's Cube, Roman, so that's definitely where my brain went.

And I was like, oh, this is impossible.

And then I realized that three by three by three.

Yeah.

It's 27.

Yep.

It's an odd number.

Odd number.

So I can't get an even number of cubes.

Can't divide that by two.

Then you were like, well, just do an even number of

doy.

So I was like, yeah, of course.

So I did a four by four.

I don't think I use that tone of voice.

No, you didn't.

That was me.

That was me being like silly with myself.

I told you as we were doing this last night, I said, tomorrow when we're in the office recording the episode, I both have a lot of dice in the office.

Yes.

I can see three different containers or bags that are full of dice.

Including a jar that

you're guessing later.

I'm sure you're not allowed near.

I've also got like cubes that link together.

Would you like a bunch of cubes that link together?

Yes, please.

Wait there.

They're on my desk.

While Matt gets those, I'll explain what I did.

So I realized that 4x4x4x4 is 64, which means that then I need to make half a cube that looks like stairs that consists of 32 cubes.

If I could build up something that looks like two sets of stairs that uses 32 of each and they're identical but then matched together, then I've solved this.

For the listeners at home, these are all the cubes that I used.

I did a video about how you can find all the 3D nets of the 4D hypercube.

And then I showed how they can tile to fill space.

And so basically got big chunks of those that were left over.

That she's going to pull some cubes off.

I sat down and I realized that

if you think about it, like from a square point of view, is fine, because you're basically going to cut one side all the way through.

And you end up with a prism of whatever your staircase looks like.

So I realized that the issue is that you you instinctively want to put one of the corners of the stairs right in the center, but that doesn't work.

And so I thought, well, if that doesn't work, and hypothetically, if there is a way to do this, I need need to do it the other way.

And so I realized the other way was to put the center right in the middle of one of the steps, like halfway across the flat.

And so I drew a square,

I put a dot in the middle for the center, and I'm going to resketch this now so we can all see it.

And I then drew the middle flat first, a third of the way across.

And then I just kind of joined up the rest.

So I filled that in like that, and I filled that in like that.

And there you are.

That's two identical halves.

So you have drawn a picture of the first step starts about about a quarter of a way up

yeah and then

so that's sort of if that first bit was a cube like that first square that first step could be a cube from the floor up yes but then i would say the next one is like two cubes across they're each a quarter yeah height and they're a third of the length of the cube across.

So that each of these runs is a third and each of these heights is up as a quarter.

My brain is just calculating.

Tell you what.

Let me do another sketch.

Let me know if this is better.

Because I would say that isn't that essentially one and then that's two and then that's one.

If you were to draw this as a.

Oh yeah, but I've just drawn that badly.

They can all be the same.

They can all be a third.

Yeah, yeah, yeah, yeah, yeah.

It's just longer steps.

Yeah, it looks like an IKEA manual.

There you are.

Get a friend and make that out of a cube.

Yeah.

So the short answer is yes.

Yes.

And this method would generalize.

You could do this for more steps.

So if you wanted more and smaller steps, you'd split it in one direction up into an odd number of pieces, in the other direction into an even number, and then you would zigzag your way up.

Yeah.

Yeah.

If you want tiny steps for tiny feet.

Yeah.

And if you want mine, which I'm still building,

finish that afterwards, and then we'll take a picture and we'll put it on socials.

But my version,

to be fair, in all honesty, I would go with yours, Matt.

They look more like traditional stairs.

They'd be much easier to use, a lot less chance of falling around.

But if you want to get all rancher,

my version sort of goes from

a corner more theatrical, if I may.

It is more theatrical, and you could put them together to make almost like a half a pyramid

type set of stairs if you wanted to sit them up against each other.

Well, I want you to finish yours.

I think you should do both halves.

I want to see them.

I'm going to.

I'm going to.

But I'm going to to wait till we take a break so that

people can go.

Ah, right.

Okay.

Yeah.

Yeah.

We'll have a break.

You can do that.

Yeah.

Yeah.

But in the magical listening land, it's done.

Oh, my goodness.

This is amazing.

If you want to see a photo of it, go to A Problem Squared on Instagram or on Twitter.

So I'm prepared to say that yours are way more theatrical.

I love them.

Thank you.

Mine are deeply boring and functional.

But I love them.

So, yeah.

But between us.

We have shown it it is possible.

So the problem, technically, is, is it really impossible for the stairs to be identical?

We've managed to show, no, it's not impossible.

And it's proof by doing one version each.

Yeah.

Proof by sticking cubes together.

Love it.

The end.

Yeah, and I want to see a photo of these stairs on stage.

Oh, yeah.

Yeah.

Jelly, let us know what you end up doing.

Yeah, yeah.

If you end up making them, take a photo of it during the play.

No rush.

Sometime in the next couple years, we want a photo.

Yeah.

Upload it.

Send us a link on the problem posing page.

And just like Pi, we're at any other business.

There's always more business in Pi.

Because it starts with three letters, A-O-B.

That one.

And has a point.

A point, which is one, four.

No, I'm trying real hard.

I like what you're doing.

Thanks.

Yeah, I'm getting there.

But let's just get into it.

What's our any other business?

You pulled out a package and now I'm excited about it.

I've pulled out a package.

Yeah, not a euphemism.

Would you like to would you like to open this?

Oh, you know what?

I'm going to open it and I'm going to reveal the contents.

Oh, that's so mean.

I know.

I dangled it in front of you and I took it away.

I know.

I just.

You're like, do you want to get the button?

Oh, I'm going to get it instead.

You're the guy in the lift no one likes.

Ready?

Yep.

Oh, I can't.

I'm going to describe for the listeners what I have just removed from the back.

Matt has just pulled out his book that he has not finished writing yet.

Correct.

It raises a lot of questions.

Yes, but you have pulled out what I imagine the book is exactly going to look like once it is available.

You've correctly worked out this is not the book.

First out, the back is completely blank.

Okay, yeah.

I didn't seen this before.

But this seems to be like a cover.

Yep.

And the inside.

Is this a proof copy of your book?

No, it's...

So the inside is someone else's book.

It's The Trading Game by Gary Stevenson.

Amazing.

Also published by Alan Lane.

So the same imprint at Penguin as me.

Does this mean that someone else is getting your book

with their cover?

Do I pay it forward?

So my people at Penguin got in touch and said, look, we've made a mock-up version of your book for photography reasons because they're doing promo stuff.

And they're like,

do you want it?

And I'm like, yeah.

Of course, yeah.

So there it is.

So they posted me someone else's book.

with a mock-up of what the cover of my book will look like on it.

Yes, I love it.

It's a great, it's a very eye-catching.

I quite like it.

It's a good cover.

This is the UK cover for people.

There's a different cover in the US.

So your third book's going to be called Love Triangle.

Love Triangle.

Perfect.

It's the third book.

Pre-sales have just started, which is very exciting.

Yes.

If people want to pre-order the book, it'll be out in the UK on the 20th of June and out in the US on the 20th of August.

August this year, 2024.

And I can see that the subtitle of your book is The Life-Changing Magic of Trigonometry.

I've got to say, I'm very sad that they didn't go with your original choice for a subtitle.

Well, getting tricky with it.

Yeah.

Yeah, yeah, yeah.

You can only make so many great suggestions

before you just got to go with what they got.

So very excited about this, and we will put some links in the description.

So if you pre-order it, that really helps me.

But if you can't afford the hardback, or if you pre-order it through my website, it's full price, but I sign it.

And you will probably still get a limited edition dust jacket.

If you pre-order it somewhere else, it's cheaper, particularly if you're in North America, because postage for me to send it to you from the UK is quite expensive.

So we will link to a variety of places where you can pre-order it.

Yes.

Down below.

Yes.

Including Waterstones, who they got in touch.

My publisher went to them and said, hey, could you offer a discount for Matt's listeners and viewers and fans?

Yeah.

And they're like, oh, yeah, sure.

We can do up to 20% off.

And I said, well, 21 is a triangle number.

So if you made it 21% off.

You're the guy where someone at the other end is getting the emails going,

so true.

This guy.

This guy.

They're like, fine.

21% off.

What code would you like?

You now realize there's going to be other people like, oh, my Matt Poco got 21%.

I know, I know.

But I had a good maths reason for it.

And they're like, what code?

You want, you know, like a triangle discount or, you know, stand-up maths, whatever.

And I'm like, if I'm allowed plus and equal signs,

the code is one plus two plus three plus four plus five plus six equals twenty one so it's a triangle number yes so so if you pre-order on waterstones uk edition 21 off if you pre-order through me 0% off but assigned but i will sign you'll have a signed first edition yeah with a limited edition dust jacket that i will give you for free so the choice is yours or wait for it to come out and paper back it'll be much cheaper then yes that's my any other business any other literal business i'm doing business you are yeah closing deals making sales nice one well

let's bring you down a notch.

Because we got a response to the poll that we did for our Californian episode.

I haven't seen it yet, but I suspect it's going to remind me why I hate democracy so much.

So we did an LA episode, and our question in that was about the Hollywood stars.

Yes.

And if you were to follow that in alphabetical order,

then what distance would you cover sort of going in between

all the stars?

And you went, okay, I'll go for the first name because some things weren't names, some things are like.

I went by whatever string of characters was on the star.

Yes.

And some,

I think it was three, three of them started with punctuation.

Yeah, like an asterisk for um sync.

Yeah.

Like a quote mark for widow.

Yes.

Yeah.

And I included, if they put the punctuation on the star, I used it as part of the text string that I then sorted you were like how does Excel sort it if Excel because I was like how do you know if an asterisk comes before a quotation mark and you were like that's how Excel does it yep I thought maybe it's to do with the QWERTY keyboard that's a whole other question different question my argument was that's dumb it was your whole argument that's a good point

if I say weird out whenever I say weird al Jankovic I'm not doing quotation marks with my fingers every time I say it like it is just a given and so I was like weird al Yankovic should just be listed

under W for weird L.

I'm having a look.

And we're in a poll.

Oh, dear.

That is decisive.

Yeah.

You know what's, I don't know if this is better or worse, because if it was like 98% against me and 2% for, then I'm very clearly in the wrong.

Yeah.

Totally misguided.

The results, if I may reveal them, are against me, 64.8%,

with me, 35.2.

So, so I'm wrong, according to the people, but totally wrong.

Like one in three people

is an idiot.

Yeah.

Though I can't believe two-thirds of you would disrespect weird owl like this.

I think you're disrespecting weird owl.

Why by using every character in their name?

By suggesting that weird owl's first the first letter of weird owl's name is a quote.

Okay, okay, okay.

I mean, I can't argue.

The results are in.

Yeah.

Wow.

Which does bring me on to some other, any other business about the Hollywood stars.

Yep.

Alan wrote into the problem posing page, selected solution in the drop down, and said, a possible explanation for the incongruence between the, I just had to make sure I was pronouncing that right, between the experimental Hollywood star distance and theoretical distance is that certain names would be disproportionately more common within certain generations who would all age up and get stars at similar times.

Okay.

Which is similar to our, I believe it was the last episode where we talked about names and trends with names and everything.

So it's funny how these sort of linked.

For example, a bunch of Michaels born in 1970 might all grow up to be famous during a certain prime, become famous enough for a star window.

Therefore...

put in together.

Exactly.

Or a small but significant amount of Michaels would all get stars at a similar time and be nearer to each other.

Interesting.

It might be interesting to look at the distribution of dates they were installed and where yeah and if the distances between them they clump some of them clump

so I thought that was really interesting

another thing I noticed pop up on the a problem squared Reddit oh yeah which I occasionally have a little

a little lurking

yeah

was that Chad Chad got a shout out in the episode said I'm so excited now I need to look up Charles Butterworth because we mentioned that from the Hollywood stars.

Yeah, everyone's favourite person.

Charles Sorry to work.

We were like, oh, yeah,

he probably looks like

a butler or something like that.

Chad did a little deep dive into it.

Said Reddit never disappoints and then has linked to another Reddit thread where it quotes, he is literally the inspiration for the beloved naval serial mascot, Captain Crunch.

Wow.

Yep.

Good work, Chad.

If you're on Reddit.

Subreddit crew.

Yeah.

My last bit of any other business?

Oh, yes.

More business.

More US.

More US.

Putting the US into business.

There is no US in business, is there?

B US.

Oh, it's right at the top.

It's right at the top.

I knew it.

Brain.

I doubted it again.

So I'm going to the US.

The reason I'm going is there's a big solar eclipse.

And my wife is a solar physicist.

So off we go.

And while I'm there, we were going to drop by New York, see some friends.

And I decided to put on an evening of a necessary detail on April 14th.

It's a Sunday.

I'll be there.

Grant Sanderson from Through Blue One Brown.

Yeah.

I'll be back.

Emma Haruka Awau, who is the current world record holder at the time of record for the number of digits of Pi ever calculated by any means.

She commanded a lot of Google Cloud server time because she works there to calculate, oh goodness, I think we're up on 100 trillion digits now.

Wow.

Phenomenal.

But Emma's going to come along, and I've seen her talk about how she calculated Pi to hundreds of trillions of digits before, and it's so fascinating.

And we've got a bunch of other incredible guests.

So we're at the Bell House in Brooklyn on Sunday, the 14th of April.

It'd be great to see a bunch of dinglets there.

Dingers, that's what we call you all.

Yeah.

Dinguses, wasn't it?

Dinguses.

Dingai.

It'd be great to see all you dingai.

Yeah.

I won't be there, but if anyone's in LA, I'll be

LA.

I'll be around.

I'm going to be in LA from mid-March until early May.

Yeah.

So

I don't don't have anything booked yet, but you know, it'll happen.

Keep an eye out on the socials.

I'm Beck Your Comedian and I'll announce.

Announce.

Yeah.

And before we finish up, we'd like to say a few thanks, some gratefulnesses,

appreciation

of our Patreon supporters who are the reason that we can continue making this podcast and the reason it's accessible to everyone.

So in order to do that, we choose three Patreon supporters at random.

Yep.

Using Matt's very spreadsheet of randomness.

Yes.

And then thank them.

And in this episode, those three Patreon supporters are

Yule Ian Freeman.

Our next one is I'm a cat.

There you go.

Ema cat.

Ema cat.

Dinas la sitter.

Yes, I pronounced it.

Dinas.

La sitta.

I I can see you, Dinas.

I'd also like to thank my co-host, Matt Parker.

That's me.

And our fantastic producer, who is also a lot like Pi,

in that without it, stuff wouldn't work.

Yeah.

Yep.

Everything would be all over.

Chaos.

Yeah, madness.

No circles.

No circles.

Lauren Armstrong Carter.

Thank you very much, Lauren.

And thank you to everyone else for listening.

Especially if you've told other people to listen to this podcast.

You're extra special.

Yeah, when you share it, when you've I see you, I see you on the socials tagging us and recommending us to people.

It always lightens my mood.

The podcast is like a pyramid scheme.

You get half the credit of the people you're recommending to listening.

I don't think it works that way at all.

I think you get the same amount of joy when you share it, no matter how many

steps it came down to get to you.

Oh, yeah.

So in a way, we're the ultimate get-rich screen.

I don't know how it works now.

I don't know where I was going with that.

I'm not getting rich and it's not quick.

No.

So, Beck, I just had to look up and remind myself.

This is why I'm on my phone while you were talking at the end there.

To remind myself how many dice are in that jar.

Yes.

And I've got the correct number in front of me.

Although I haven't double-checked for a while.

Maybe people have been helping themselves to the dice.

I'll have to check.

So, you recently picked 469,

to which I would have said lower.

Oh, which you would have said lower.

Yes.

And the lowest one below that is 453, to which I would have said higher.

So that's the window we're in now.

It's not a big window.

It's not.

I could get it.

This could be it.

This could be it.

I'm going to say

400 and 61.

Lower.

Oof.

Can't be long now.

Not long now.

For the hundredth episode.

Bring it in, Baby.